Data analysis using ML models with OPTIMEO

Let’s create an experimental_data(temp, conc) function that simulates the yield of a chemical reaction based on temperature, concentration A, concentration B and concentration C.

import numpy as np
import pandas as pd
import plotly.graph_objects as go
import plotly.io as pio
pio.renderers.default = "notebook"

def experimental_data(temp, cA, cB, cC):
    """
    This function simulates experimental data based on temperature and concentrations.
    The function is not based on any real experimental data and is purely for demonstration purposes.
    """
    out = .2*temp + .5*temp*cA + (cA)/3 + (1 - cB)**2/2 + (3 - cC)/1.5 + np.random.normal(0, 0.2, len(temp))
    return out

def generate_data(N=100):
    temp = np.random.uniform(0, 100, N)
    cA = np.random.uniform(0, 1, N)
    cB = np.random.uniform(0, 1, N)
    cC = np.random.uniform(0, 1, N)
    exp_response = experimental_data(temp, cA, cB, cC)
    # Create a DataFrame with the generated data
    df = pd.DataFrame({'temp': temp, 
                       'cA': cA, 
                       'cB': cB, 
                       'cC': cC, 
                       'response': exp_response})
    return df

df = generate_data(50)
df.to_csv('dataML.csv', index=False)
df.head()

	temp	cA	cB	cC	response
0	65.150929	0.856934	0.143145	0.202648	43.365583
1	70.857440	0.611890	0.588674	0.089296	37.581891
2	84.400019	0.827279	0.546844	0.081419	53.979769
3	32.449034	0.888718	0.713346	0.640800	23.005230
4	64.010182	0.757054	0.611216	0.881803	38.662838

Now, we will use the OPTIMEO package to analyse the data.

from optimeo.analysis import *

data = pd.read_csv('dataML.csv')
factors = data.columns[:-1]
response = data.columns[-1]
analysis = DataAnalysis(data, factors, response)
analysis

DataAnalysis(data=(50, 5), factors=Index(['temp', 'cA', 'cB', 'cC'], dtype='object'), response=response, model_type=None, split_size=0.2, encoders={})

First, let’s look at a simple linear model:

analysis.compute_linear_model()
analysis.linear_model.summary()

OLS Regression Results

Dep. Variable:	response	R-squared:	0.926
Model:	OLS	Adj. R-squared:	0.919
Method:	Least Squares	F-statistic:	139.9
Date:	Thu, 17 Apr 2025	Prob (F-statistic):	8.94e-25
Time:	09:43:34	Log-Likelihood:	-144.72
No. Observations:	50	AIC:	299.4
Df Residuals:	45	BIC:	309.0
Df Model:	4
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	-12.9717	2.653	-4.889	0.000	-18.316	-7.628
temp	0.4709	0.023	20.630	0.000	0.425	0.517
cA	25.9518	2.121	12.239	0.000	21.681	30.223
cB	1.6269	2.381	0.683	0.498	-3.168	6.422
cC	-0.0613	2.581	-0.024	0.981	-5.260	5.137

Omnibus:	0.893	Durbin-Watson:	1.991
Prob(Omnibus):	0.640	Jarque-Bera (JB):	0.331
Skew:	0.153	Prob(JB):	0.848
Kurtosis:	3.256	Cond. No.	324.

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

figs = analysis.plot_linear_model()
for fig in figs:
    fig.show()

The equation used for the fit is this one, you can change it if you want, e.g to add interaction terms or other polynomial terms:

analysis.write_equation()

'response ~ temp + cA + cB + cC '

analysis.equation = 'response ~ temp+ temp:cA + cA + cB + cC'
analysis.compute_linear_model()
analysis.linear_model.summary()

OLS Regression Results

Dep. Variable:	response	R-squared:	1.000
Model:	OLS	Adj. R-squared:	1.000
Method:	Least Squares	F-statistic:	4.641e+04
Date:	Thu, 17 Apr 2025	Prob (F-statistic):	1.09e-80
Time:	09:43:35	Log-Likelihood:	4.5898
No. Observations:	50	AIC:	2.820
Df Residuals:	44	BIC:	14.29
Df Model:	5
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	2.3612	0.179	13.201	0.000	2.001	2.722
temp	0.1984	0.002	83.340	0.000	0.194	0.203
temp:cA	0.5014	0.004	131.248	0.000	0.494	0.509
cA	0.1534	0.224	0.684	0.498	-0.299	0.606
cB	-0.4569	0.123	-3.728	0.001	-0.704	-0.210
cC	-0.2756	0.132	-2.092	0.042	-0.541	-0.010

Omnibus:	1.678	Durbin-Watson:	1.538
Prob(Omnibus):	0.432	Jarque-Bera (JB):	1.536
Skew:	-0.309	Prob(JB):	0.464
Kurtosis:	2.405	Cond. No.	538.

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

figs = analysis.plot_linear_model()
for fig in figs:
    fig.show()

Now let’s make a ML model to predict the yield based on the temperature and concentrations of A, B and C.

analysis.model_type = "ElasticNetCV"
# analysis.model_type = "RidgeCV"
# analysis.model_type = "LinearRegression"
# analysis.model_type = "RandomForest"
# analysis.model_type = "GaussianProcess"
# analysis.model_type = "GradientBoosting"
MLmodel = analysis.compute_ML_model()
figs = analysis.plot_ML_model()
for fig in figs:
    fig.show()

And if we want to make a prediction:

new_value = pd.DataFrame({'temp': [50], 
                          'cA': [0.35], 
                          'cB': [0.5], 
                          'cC': [0.5]})
analysis.predict(new_value)

	prediction	model
0	20.582657	ElasticNetCV
1	20.745541	Linear Model